One of the many famous theorems proven by William Thurston is that a closed connected smooth manifold $M$ admits a codimension 1 smooth foliation if and only if $\chi(M)=0$:

W.P. Thurston, Existence of codimension-one foliations. Ann. of Math. (2) 104 (1976), no. 2, 249–268.

The "only if" part of this theorem, of course, is much simpler and older.

My question is: What happens in the topological category?

I convinced myself (although I did not write down a detailed proof) that if a closed connected topological manifold $M$ admits a topological codimension 1 foliation, then $\chi(M)=0$. (The proof imitates the "standard" smooth argument, using a map from $M$ to the symmetric product of $M$ with itself, instead of constructing a line bundle on $M$ in the smooth case.) The true question is if the hard part of Thurston's theorem holds in the topological category:

Question. Suppose that $M$ is a closed connected topological manifold with $\chi(M)=0$. Does $M$ admit a codimension 1 foliation?

I could not find anything on MathSciNet using the obvious algorithm.